Some results on the lattice of closed ideals of $\mathcal L^r(X)$ for $X$ of the form $(\bigoplus_i \ell _p^i)_q$

Ariel Blanco Studia Mathematica MSC: Primary 46H10, 47L10; Secondary 46B42, 47B65. DOI: 10.4064/sm200305-25-1 Opublikowany online: 31 May 2021

Streszczenie

We study the lattice of closed (order and algebra) ideals of $\mathcal L^r(X)$ when $X$ is a Banach lattice of the form $(\bigoplus _i \ell _p^i)_q$ $(p\in [1,\infty ]$, $q\in [1,\infty )\cup \{0\} \,\&\, p\ne q)$. We show that for every such $X$, $\mathcal L^r(X)$ has a unique maximal (order and algebra) ideal. For $1 \lt p \lt \infty $ and $q\in \{0,1\}$, we show, in particular, that the lattice of closed (order and algebra) ideals of $\mathcal L^r(X)$ contains at least five distinct ideals.

Autorzy

  • Ariel BlancoMathematical Sciences Research Centre
    Queen’s University Belfast
    University Road
    BT7 1NN Belfast, UK
    e-mail

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